A001263 -id:A001263 - cp's OEIS frontend

A103367 Absolute row sums of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

1, 2, 6, 26, 160, 1372, 15974, 245142, 4817712, 118198568, 3542890648, 127417949496, 5415490994368, 268526379444104, 15363229400769566, 1004545432884250126, 74441340170270921952, 6205992783298302119536

Offset: 2

Paul D. Hanna, Feb 02 2005

nonn

The ratio of the absolute row sum to the absolute value of column 1 for row n of triangle A103364, as n grows, approaches the limit given by limit_{n->inf} Sum_{k=1,n} |A103364(n,k)| / |A103365(n)| = 4.37281937295201487280058335227924496590808747668123198019676685492873...

Cf. A001263, A103364, A103365, A103366.

{a(n)=if(n<1,0,sum(k=1,n,abs(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^-1)[n,k]))}

A103370 Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).

Original entry on oeis.org

1, 3, 12, 57, 303, 1743, 10629, 67791, 448023, 3047745, 21235140, 150969195, 1091936745, 8016114681, 59616180828, 448459155063, 3407842605039, 26131449100821, 202011445055436, 1573171285950639, 12333030718989969

Offset: 1

Paul D. Hanna, Feb 02 2005

nonn

			From _Paul D. Hanna_, Feb 01 2009: (Start)
G.f.: A(x) = 1 + 3*x + 12*x^2/3 + 57*x^3/18 + 303*x^4/180 + 1743*x^5/2700 +...+ a(n)*x^n/[n!*(n+1)!/2^n] +...
A(x) = B(x)^3 where:
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 +...+ x^n/[n!*(n+1)!/2^n] +... (End)

Vincenzo Librandi, Table of n, a(n) for n = 1..200
Jonathan M. Borwein, A short walk can be beautiful, 2015.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
J. M. Borwein, Adventures with the OEIS: Five sequences Tony may like, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
Jonathan M. Borwein, Armin Straub and Christophe Vignat, Densities of short uniform random walks, Part II: Higher dimensions, Preprint, 2015.

Cf. A000108, A001263, A008277, A095801.

RecurrenceTable[{(n + 1) * (n + 2) * a[n] == 2 * (5 * n^2 - 2) * a[n - 1] - 9 * (n - 2) * (n - 1) * a[n - 2], a[1] == 1, a[2] == 3}, a, {n, 21}] (* Vaclav Kotesovec, Oct 17 2012 *)

{a(n)=if(n<1,0,sum(k=1,n,(matrix(n,n,m,j,binomial(m-1,j-1)*binomial(m,j-1)/j)^2)[n,k]))}

{a(n)=local(B=sum(k=0,n,x^k/(k!*(k+1)!/2^k))+x*O(x^n));polcoeff(B^3,n)*n!*(n+1)!/2^n} \\ Paul D. Hanna, Feb 01 2009

G.f. satisfies: A(x) = B(x)^3 where A(x) = Sum_{n>=0} a(n)*x^n/[n!*(n+1)!/2^n] and B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n]. - Paul D. Hanna, Feb 01 2009
Recurrence: (n+1)*(n+2)*a(n) = 2*(5*n^2-2)*a(n-1) - 9*(n-2)*(n-1)*a(n-2). - Vaclav Kotesovec, Oct 17 2012
a(n) ~ 3^(2*n+5/2)/(4*Pi*n^3). - Vaclav Kotesovec, Oct 17 2012
G.f.: ((x-1)^2/(4*x*(1-9*x)^(2/3))*(-3*hypergeom([1/3, 1/3],[1],-27*x*(x-1)^2/(9*x-1)^2)+(3*x+1)^3*(9*x-1)^(-2)*hypergeom([4/3, 4/3],[2],-27*x*(x-1)^2/(9*x-1)^2)))-1+1/(2*x). - Mark van Hoeij, May 14 2013
G.f.: -(x-1)^2*hypergeom([1/3, 4/3],[2],-27*x*(x-1)^2/(9*x-1)^2)/(2*x*(1-9*x)^(2/3))-1+1/(2*x). - Mark van Hoeij, Nov 12 2023

A105556 Triangle, read by rows, such that column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263, for n>=0.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 14, 12, 4, 1, 1, 42, 57, 22, 5, 1, 1, 132, 303, 148, 35, 6, 1, 1, 429, 1743, 1144, 305, 51, 7, 1, 1, 1430, 10629, 9784, 3105, 546, 70, 8, 1, 1, 4862, 67791, 90346, 35505, 6906, 889, 92, 9, 1, 1, 16796, 448023, 885868, 444225, 99156

Offset: 0

Paul D. Hanna, Apr 14 2005

Column 1 is the Catalan numbers A000108 (offset 1).

			Triangle begins:
  1;
  1,    1;
  1,    2,     1;
  1,    5,     3,     1;
  1,   14,    12,     4,     1;
  1,   42,    57,    22,     5,    1;
  1,  132,   303,   148,    35,    6,   1;
  1,  429,  1743,  1144,   305,   51,   7,  1;
  1, 1430, 10629,  9784,  3105,  546,  70,  8, 1;
  1, 4862, 67791, 90346, 35505, 6906, 889, 92, 9, 1;
  ...
From _Paul D. Hanna_, Feb 01 2009: (Start)
G.f. for rows n=0..3 are:
B(x) = 1 + x + x^2/3 + x^3/18 + x^4/180 + x^5/2700 + ... + x^n/[n!*(n+1)!/2^n] + ...
B(x)^2 = 1 + 2*x + 5*x^2/3 + 14*x^3/18 + 42*x^4/180 + ... + A000108(n)*x^n/[n!*(n+1)!/2^n] + ...
B(x)^3 = 1 + 3*x +12*x^2/3 + 57*x^3/18 +303*x^4/180 + ... + A103370(n)*x^n/[n!*(n+1)!/2^n] + ...
B(x)^4 = 1 + 4*x +22*x^2/3 +148*x^3/18+1144*x^4/180 + 9784*x^5/2700 + 90346*x^5/56700 + ... (End)

Cf. A001263, A105557 (row sums), A103370 (column 2).

Cf. A155926. - Paul D. Hanna, Feb 01 2009

{T(n,k)=local(N=matrix(n+1,n+1,m,j,if(m>=j, binomial(m-1,j-1)*binomial(m,j-1)/j))); sum(j=0,n-k,(N^k)[n-k+1,j+1])}

{T(n,k)=local(B=sum(j=0,n-k,x^j/(j!*(j+1)!/2^j))+x*O(x^(n-k))); polcoeff(B^(k+1),n-k)*(n-k)!*(n-k+1)!/2^(n-k)} \\ Paul D. Hanna, Feb 01 2009

From Paul D. Hanna, Feb 01 2009: (Start)
G.f. of column k = B(x)^(k+1) where B(x) = Sum_{n>=0} x^n/[n!*(n+1)!/2^n];
T(n,K) = [x^(n-k)] B(x)^(k+1) * (n-k)!*(n-k+1)!/2^(n-k) for n >= k >= 0. (End)

A132787 Triangle read by rows: T(n,k) = 2*A001263(n,k) - 1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 11, 11, 1, 1, 19, 39, 19, 1, 1, 29, 99, 99, 29, 1, 1, 41, 209, 349, 209, 41, 1, 1, 55, 391, 979, 979, 391, 55, 1, 1, 71, 671, 2351, 3527, 2351, 671, 71, 1, 1, 89, 1079, 5039, 10583, 10583, 5039, 1079, 89, 1, 1, 109, 1649, 9899, 27719, 38807, 27719, 9899, 1649, 109, 1

Offset: 1

Gary W. Adamson, Aug 30 2007

			First few rows of the triangle are:
  1;
  1, 1;
  1, 5, 1;
  1, 11, 11, 1;
  1, 19, 39, 19, 1;
  1, 29, 99, 99, 29, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Column 2 is A018387.
Row sums are A132788.
Cf. A001263.

T(n,k) = if(k<=n, 2*binomial(n-1, k-1) * binomial(n, k-1) / k - 1, 0); \\ Andrew Howroyd, Aug 10 2018

Equals 2*A001263 - A000012 as infinite lower triangular matrices; where A001263 = the Narayana triangle.

T(n,k) = 2*binomial(n-1, k-1) * binomial(n, k-1) / k - 1. - Andrew Howroyd, Aug 10 2018

a(20) corrected and terms a(56) and beyond from Andrew Howroyd, Aug 10 2018

A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices.

Original entry on oeis.org

1, 3, 3, 6, 18, 6, 10, 60, 60, 10, 15, 150, 300, 150, 15, 21, 315, 1050, 1050, 315, 21, 28, 588, 2940, 4900, 2940, 588, 28, 36, 1008, 7056, 17640, 17640, 7056, 1008, 36, 45, 1620, 15120, 52920, 79380, 52920, 15120, 1620, 45, 55, 2475, 29700, 138600, 291060

Offset: 1

Gary W. Adamson, Sep 02 2007

			First few rows of the triangle are:
1;
3, 3;
6, 18, 6;
10, 60, 60, 10;
15, 150, 300, 150, 15;
21, 315, 1050, 1050, 315, 21;
...

Cf. A127773, A001263, A002457 (row sums), A006472. A002695, A008459.

A132818 := proc(n,k)
    (n+1-k)*binomial(n+1,k)*binomial(n,k-1)/2 ;
end proc: # R. J. Mathar, Jul 29 2015

T(n,k) = A000217(n) * A001263(n,k).
Let a(n) = A006472(n), the 'triangular' factorial numbers. Then a(n)/(a(k)*a(n-k)) produces the present triangle (with a different offset). - Peter Bala, Dec 07 2011
T(n,k) = 1/2*(n+1-k)*C(n+1,k)*C(n,k-1), for n,k >= 1. O.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(3/2) = x*y + x^2*(3*y + 3*y^2) + x^3*(6*y + 18*y^2 + 6*y^3) + .... Cf. A008459 with o.g.f.: x*y/((1-x-x*y)^2 - 4*x^2*y)^(1/2). Sum {k = 1..n-1} T(n,k)*2^(n-k) = A002695(n). - Peter Bala, Apr 10 2012

Corrected by R. J. Mathar, Jul 29 2015

A375999 Narayana numbers (A001263), sorted, duplicates removed.

Original entry on oeis.org

1, 3, 6, 10, 15, 20, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 175, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A001263, A006987, A024412, A376000, A376001.

from bisect import insort
from itertools import islice
def A375999_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        yield N0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0: break
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if n == 2*k-1:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A375999_list(nmax):
    return list(islice(A375999_generator(),nmax))

A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

Original entry on oeis.org

1, 6, 10, 15, 21, 28, 36, 45, 50, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 196, 210, 231, 253, 276, 300, 325, 336, 351, 378, 406, 435, 465, 490, 496, 528, 540, 561, 595, 630, 666, 703, 741, 780, 820, 825, 861, 903, 946, 990, 1035, 1081, 1128, 1176, 1210

Offset: 1

Pontus von Brömssen, Sep 06 2024

nonn

All Narayana numbers A001263(n,k) with n != 2*k-1, are terms since A001263(n,k) = A001263(n,n+1-k). In particular, all positive triangular numbers except 3 are terms. Are there any other terms, i.e., is there a number A001263(2*k-1,k), k >= 2, that can be written as a Narayana number in another way? Any such number would also be a term of A376001.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A000217, A001263, A006987, A098564, A374796, A375572, A375999, A376001.

from bisect import insort
from itertools import islice
def A376000_generator():
    yield 1
    nkN_list = [(3, 2, 3)] # List of triples (n, k, A001263(n, k)), sorted by the last element.
    while 1:
        N0 = nkN_list[0][2]
        c = 0
        while 1:
            n, k, N = nkN_list[0]
            if N > N0:
                if c >= 2: yield N0
                break
            central = n==2*k-1
            c += 2-central
            del nkN_list[0]
            insort(nkN_list, (n+1, k, n*(n+1)*N//((n-k+1)*(n-k+2))), key=lambda x:x[2])
            if central:
                insort(nkN_list, (n+2, k+1, 4*n*(n+2)*N//(k+1)**2), key=lambda x:x[2])
def A376000_list(nmax):
    return list(islice(A376000_generator(),nmax))

A102812 a(0) = 1, a(n) = Sum_{k=1..n} A001263(n,k)*a(k-1) where A001263(n,k) are Narayana numbers.

Original entry on oeis.org

1, 1, 2, 6, 25, 136, 927, 7690, 75913, 876026, 11649009, 176389234, 3011057742, 57453864447, 1216376055329, 28390144638804, 726364328468535, 20269424489285300, 614148655197018345, 20123000681064168210, 710410428812211243412, 26932611130467463342807

Offset: 0

Gerald McGarvey, Feb 26 2005

nonn

Eigensequence of A001263. - Philippe Deléham, Jun 22 2007

Andrew Howroyd, Table of n, a(n) for n = 0..200

Cf. A001263.

\\ here T(n,k) is A001263.
T(n, k) ={if(k==0, 0, binomial(n-1, k-1) * binomial(n, k-1) / k)}
seq(n)={my(a=vector(n+1)); a[1]=1; for(n=1, n, a[n+1]=sum(k=1, n, a[k]*T(n,k))); a} \\ Andrew Howroyd, Nov 06 2019

Terms a(12) and beyond from Andrew Howroyd, Nov 06 2019

A104711 Triangle T(n,m) = sum_{k=m..n} A001263(k,m).

Original entry on oeis.org

1, 2, 1, 3, 4, 1, 4, 10, 7, 1, 5, 20, 27, 11, 1, 6, 35, 77, 61, 16, 1, 7, 56, 182, 236, 121, 22, 1, 8, 84, 378, 726, 611, 218, 29, 1, 9, 120, 714, 1902, 2375, 1394, 365, 37, 1, 10, 165, 1254, 4422, 7667, 6686, 2885, 577, 46, 1, 11, 220, 2079, 9372, 21527, 26090, 16745

Offset: 1

Gary W. Adamson, Mar 19 2005

This summation over columns of the Narayana triangle could also be defined as a multiplication

of the Narayana triangle from the left by the lower-left triangle represented by the all-1 sequence A000012.

			First few rows of the triangle are:
1;
2, 1;
3, 4, 1;
4, 10, 7, 1;
5, 20, 27, 11, 1;
6, 35, 77, 61, 16, 1;
...

Cf. A014138, A104710, A005585, A000124.

from sympy import binomial
def A001263(n,m):
    return binomial(n-1,m-1)*binomial(n,m-1)//m
def A104711(n,m):
    a = 0
    for k in range(m,n+1):
        a += A001263(k,m)
    return a
print([A104711(n,m) for n in range(20) for m in range(1,n+1)]) # R. J. Mathar, Oct 11 2009

Row sums: sum_{m=1..n} T(n,m) = A014138(n).

Extended by R. J. Mathar, Oct 11 2009

A118976 Triangle read by rows: T(n,k) = binomial(n-1,k-1)binomial(n,k-1)/k + binomial(n-1,k)binomial(n,k)/(k+1) (1 <= k <= n). In other words, to each entry of the Narayana triangle (A001263) add the entry on its right.

Original entry on oeis.org

1, 2, 1, 4, 4, 1, 7, 12, 7, 1, 11, 30, 30, 11, 1, 16, 65, 100, 65, 16, 1, 22, 126, 280, 280, 126, 22, 1, 29, 224, 686, 980, 686, 224, 29, 1, 37, 372, 1512, 2940, 2940, 1512, 372, 37, 1, 46, 585, 3060, 7812, 10584, 7812, 3060, 585, 46, 1, 56, 880, 5775, 18810, 33264, 33264, 18810, 5775, 880, 56, 1

Offset: 1

Gary W. Adamson, May 07 2006

Sum of entries in row n = 2*Cat(n)-1, where Cat(n) are the Catalan numbers (A000108).

Row sums = A131428 starting (1, 3, 9, 27, 83, ...). - Gary W. Adamson, Aug 31 2007

			First few rows of the triangle:
   1;
   2,  1;
   4,  4,   1;
   7, 12,   7,  1;
  11, 30,  30, 11,  1;
  16, 65, 100, 65, 16, 1;
...
Row 4 of the triangle = (7, 12, 7, 1), derived from row 4 of the Narayana triangle, (1, 6, 6, 1): = ((1+6), (6+6), (6+1), (1)).

G. C. Greubel, Rows n = 1..100 of triangle, flattened

Cf. A001263, A034856, A000108, A131428.

B:=Binomial; Flat(List([1..12], n-> List([1..n], k-> B(n-1,k-1)*B(n,k-1)/k + B(n-1,k)*B(n,k)/(k+1) ))); # G. C. Greubel, Aug 12 2019

B:=Binomial; [B(n-1,k-1)*B(n,k-1)/k + B(n-1,k)*B(n,k)/(k+1): k in [1..n], n in [1..12]]; // G. C. Greubel, Aug 12 2019

T:=(n,k)->binomial(n-1,k-1)*binomial(n,k-1)/k+binomial(n-1,k) *binomial(n,k)/ (k+1): for n from 1 to 12 do seq(T(n,k),k=1..n) od; # yields sequence in triangular form
# Alternatively:
gf := 1 - ((1/2)*(x + 1)*(sqrt((x*y + y - 1)^2 - 4*y^2*x) + x*y + y - 1))/(y*x):
sery := series(gf, y, 10): coeffy := n -> expand(coeff(sery, y, n)):
seq(print(seq(coeff(coeffy(n), x, k), k=1..n)), n=1..8); # Peter Luschny, Oct 21 2020

With[{B=Binomial}, Table[B[n-1,k-1]*B[n,k-1]/k + B[n-1,k]*B[n,k]/(k+1), {n,12}, {k,n}]//Flatten] (* G. C. Greubel, Aug 12 2019 *)

T(n,k) = b=binomial; b(n-1,k-1)*b(n,k-1)/k + b(n-1,k)*b(n,k)/(k+1);
for(n=1,12, for(k=1,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Aug 12 2019

def T(n, k):
    b=binomial
    return b(n-1,k-1)*b(n,k-1)/k + b(n-1,k)*b(n,k)/(k+1)
[[T(n, k) for k in (1..n)] for n in (1..12)] # G. C. Greubel, Aug 12 2019

G.f.: A001263(x, y)*(x + x*y) + x*y. - Vladimir Kruchinin, Oct 21 2020

Edited by N. J. A. Sloane, Nov 29 2006

cp's OEIS Frontend

A103367 Absolute row sums of triangle A103364, which equals the matrix inverse of the Narayana triangle (A001263).

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A103370 Row sums of triangle A095801 (matrix square of the Narayana triangle A001263).

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Mathematica

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A105556 Triangle, read by rows, such that column n equals the row sums of A001263^n, which is the n-th matrix power of the Narayana triangle A001263, for n>=0.

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A132787 Triangle read by rows: T(n,k) = 2*A001263(n,k) - 1.

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Extensions

A132818 The matrix product A127773 * A001263 of infinite lower triangular matrices.

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A375999 Narayana numbers (A001263), sorted, duplicates removed.

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A376000 Numbers that can be written as a Narayana number (A001263) in at least 2 ways.

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A102812 a(0) = 1, a(n) = Sum_{k=1..n} A001263(n,k)*a(k-1) where A001263(n,k) are Narayana numbers.

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A118976 Triangle read by rows: T(n,k) = binomial(n-1,k-1)binomial(n,k-1)/k + binomial(n-1,k)binomial(n,k)/(k+1) (1 <= k <= n). In other words, to each entry of the Narayana triangle (A001263) add the entry on its right.